Equivariant Discrete Normalizing Flows

At its core, generative modeling seeks to uncover the underlying factors that give rise to observed data that can often be modelled as the natural symmetries that manifest themselves through invariances and equivariances to certain transformations laws. However, current approaches are couched in the formalism of continuous normalizing flows that require the construction of equivariant vector fields -- inhibiting their simple application to conventional higher dimensional generative modelling domains like natural images. In this paper we focus on building equivariant normalizing flows using discrete layers. We first theoretically prove the existence of an equivariant map for compact groups whose actions are on compact spaces. We further introduce two new equivariant flows: $G$-coupling Flows and $G$-Residual Flows that elevate classical Coupling and Residual Flows with equivariant maps to a prescribed group $G$. Our construction of $G$-Residual Flows are also universal, in the sense that we prove an $G$-equivariant diffeomorphism can be exactly mapped by a $G$-residual flow. Finally, we complement our theoretical insights with experiments -- for the first time -- on image datasets like CIFAR-10 and show $G$-Equivariant Discrete Normalizing flows lead to increased data efficiency, faster convergence, and improved likelihood estimates.